the boss checked my computer history and saw AF

[Athene]

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You probably have a fair shot at collecting unemployment, fuzzy; Especially if there wasn't a pattern of reprimands/write-ups. It wouldn't be much, but you wouldn't have to blow through your savings quite as fast or be without income at all. It definitely decreases the odds of any passive negativity directed towards you when prospective employers inquire into your job history. Plus, you'd just be sticking it to your former employer in general.

Best of luck to you. Hope something turns up soon.

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Yeah, Thena . . . thanks . . . I'll get busy on that Monday.  The HR director promised me that if I applied for unemployment, she would approve it, despite the fact that I resigned.  (Sometimes a resignation will make you ineligible for unemployment.)  
   I'm busy sending out job applications and resumes.  I'll keep you guys updated!

I wasn't aware that they'd 'let' you resign. Assholes.

I was always under the impression that resigning makes collecting unemployment impossible, but it probably works more towards your favor during future job searches, at any rate. 

If HR is somehow willing able to make good on their promise, great. 
If not, I suppose it would be best to just drop the matter and move on, as you mentioned earlier.